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Companion matrices of the second type are characterized by properties that involve bilinear maps.
Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear…
A ribbon is a double structure on P^1. The geometry of a ribbon is closely related to that of a smooth curve. In this note we consider linear series on ribbons. Our main result is an explicit determinantal description for the locus…
The ability of neural networks to represent more features than neurons makes interpreting them challenging. This phenomenon, known as superposition, has spurred efforts to find architectures that are more interpretable than standard…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer…
Several primal and dual characterizations of regularity properties of collections of sets in normed linear spaces are discussed. Relationships between regularity properties of collections of sets and those of set-valued mappings are…
Trilinear mappings appear naturally when performing spatial isogeometric discretizations of degree $p = 1$. Among them, birational maps are characterized by the property that both the mapping and the associated inverse map are rational and…
We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces.
In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in…
Sets in R^n in which every pair of elements x, y can be connected by a path in the set of length bounded by a constant multiple of the distance between x and y are considered.
We prove bilinear inequalities for differential operators in $\mathbb{R}^2$. Such type inequalities turned out to be useful for anisotropic embedding theorems for overdetermined systems and the limiting order summation exponent. However,…
We prove that there is a structure, indeed a linear ordering, whose degree spectrum is the set of all non-hyperarithmetic degrees. We also show that degree spectra can distinguish measure from category.
We say that two elements of a group or semigroup are $\Bbbk$-linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$-linear conjugacy for finite semigroups…
Let $B$ be a bilinear form on pairs of points in the complex plane, of the form $B(p,q) = p^TMq$, for an invertible $2\times2$ complex matrix $M$. We prove that any finite set $S$ contained in an irreducible algebraic curve $C$ of degree…
We completely characterize definable linear orders in o-minimal structures expanding groups. For example, let (P,<_p) be a linear order definable in the real field R. Then (P,<_p) embeds definably in (R^{n+1},<_l), where <_l is the…
Recently, linear codes constructed by defining sets have attracted a lot of study, and many optimal linear codes with a few weights have been produced. The objective of this paper is to present a class of binary linear codes with three…
Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…
Large sets of equiangular lines are constructed from sets of mutually unbiased bases, over both the complex and the real numbers.